Global Sobolev regularity for general elliptic equations of $p$-Laplacian type

Abstract

We derive global gradient estimates for $W_0^{1,p}(\Omega )$-weak solutions to quasilinear elliptic equations of the form $$\mathrm{div}\mathbf{a}(x,u,Du)=\mathrm{div}(|F{|}^{p-2}F)$$ over $n$-dimensional Reifenberg flat domains. The nonlinear term of the elliptic differential operator is supposed to be small-BMO with respect to $x$ and Hölder continuous in $u.$ In the case when $p\ge n,$ we allow only continuous nonlinearity in $u.$ Our result highly improves the known regularity results available in the literature. In fact, we are able not only to weaken the regularity requirement on the nonlinearity in $u$ from Lipschitz continuity to Hölder one, but we also find a very lower level of geometric assumptions on the boundary of the domain to ensure global character of the obtained gradient estimates.